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利用Steinmetz固体求球体积

利用Steinmetz固体求球体积
几何画板及绘图内容加载中，请稍等……
2013-11-12 11:43
函数立体几何微积分几何作图
①本范例演示如何通过构造Steinmetz固体来推导球体积公式；②涉及的核心数学概念包括体积计算、立体几何、积分思想；③作图要点——使用了Sphere（球）、Polygon（多边形）、Prism（棱柱）、Sequence（数列）、Midpoint（中点）、Plane（平面）等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者学习体积推导方法。

GeoGebra 命令及参数

  1. Sphere(A, 1) → a
  2. Polygon(B, C, D, E) → poly1, b, c, d, e
  3. Prism(poly1, F) → prism1, G, edgeCG, edgeDH, edgeEI, edgeFG, edgeGH, edgeHI, edgeFI, H, I, faceBCGF, faceCDHG, faceDEIH, faceBEIF, faceFGHI, edgeBF
  4. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L1
  5. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L2
  6. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L3
  7. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L4
  8. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L5
  9. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L6
  10. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L7
  11. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L8
  12. Midpoint(E, B) → M1
  13. Midpoint(C, D) → M2
  14. Midpoint(G, H) → M3
  15. Midpoint(F, I) → M4
  16. Midpoint(B, C) → M5
  17. Midpoint(E, D) → M6
  18. Midpoint(I, H) → M7
  19. Midpoint(F, G) → M8
  20. Midpoint(H, D) → M9
  21. Midpoint(G, C) → M10
  22. Midpoint(F, B) → M11
  23. Midpoint(I, E) → M12
  24. Plane(M1, M2, M3) → i
  25. Plane(M5, M6, M7) → j
  26. Plane(M9, M10, M11) → k
  27. Sequence((0, sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L9
  28. Sequence((sqrt(1 - k^(2)), 0, k + 1), k, 0, 1, 0.01) → L10
  29. Sequence((0, 0, k + 1), k, 0, 1, 0.01) → L11
  30. Sequence((0 + a2, 1 + a2, k + 1), k, 0, 1, 0.01) → L12
  31. Sequence((1 + a2, 1 + a2, k + 1), k, 0, 1, 0.01) → L13
  32. Sequence((1 + a2, 0 + a2, k + 1), k, 0, 1, 0.01) → L14
  33. Sequence((sqrt(1 - k^(2)) + a2, 0 + a2, k + 1), k, 0, 1, 0.01) → L15
  34. Sequence((sqrt(1 - k^(2)) + a2, sqrt(1 - k^(2)) + a2, k + 1), k, 0, 1, 0.01) → L16
  35. Sequence((0 + a2, sqrt(1 - k^(2)) + a2, k + 1), k, 0, 1, 0.01) → L17
  36. Sequence(Polygon[Element[L12, i], Element[L13, i], Element[L14, i], Element[L15, i], Element[L16, i], Element[L17, i]], i, 1, 101, 1) → list4
  37. Sequence(Polygon[Element[L1, i], Element[L2, i], Element[L3, i], Element[L4, i]], i, 1, 101, 1) → list1
  38. Sequence(Polygon[Element[L5, i], Element[L6, i], Element[L7, i], Element[L8, i]], i, 1, 101, 1) → list2
  39. Sequence(Polygon[Element[L1, i], Element[L2, i], Element[L3, i], Element[L4, i]], i, 1, 101, 104 - (3 * a1)) → list5
  40. Sequence(Polygon[Element[L5, i], Element[L6, i], Element[L7, i], Element[L8, i]], i, 1, 101, 104 - (3 * a1)) → list6
  41. Sequence(Polygon[Element[L11, i], Element[L9, i], Element[L1, i], Element[L10, i]], i, 1, 101, 1) → list3
  42. Polygon(Element[L12, a3], Element[L13, a3], Element[L14, a3], Element[L15, a3], Element[L16, a3], Element[L17, a3]) → poly2, q, r, s_1, t, a_1, b_1
  43. Sequence((1 + a2 - 1.5, 1 + a2 - 1.5, k + 1), k, 0, 1, 0.01) → L18
  44. Sequence(((-k) + a2 - 0.5, 0 + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L19
  45. Sequence(((-k) + a2 - 0.5, (-k) + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L20
  46. Sequence((0 + a2 - 0.5, (-k) + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L21
  47. Sequence(Polygon[Element[L18, i], Element[L19, i], Element[L20, i], Element[L21, i]], i, 1, 101, 1) → list7
  48. Polygon(Element[L18, a3], Element[L19, a3], Element[L20, a3], Element[L21, a3]) → poly3, c_1, d_1, e_1, f_1
  49. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L1_1
  50. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L2_1
  51. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L3_1
  52. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L4_1
  53. Sequence(Polygon[Element[L1_1, i], Element[L2_1, i], Element[L3_1, i], Element[L4_1, i]], i, 1, 101, 1) → list1_1
  54. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L5_1
  55. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L6_1
  56. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L7_1
  57. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L8_1
  58. Sequence(Polygon[Element[L5_1, i], Element[L6_1, i], Element[L7_1, i], Element[L8_1, i]], i, 1, 101, 1) → list2_1

AI 绘图提示语参考

画一个球体，并展示如何通过构建施泰纳固体来推导球体体积公式。首先绘制一个单位球体，然后创建一个多边形并生成其棱柱。接着，利用参数方程绘制多个四边形，并通过这些四边形构建多个平面。最后，将这些平面与棱柱结合，形成一个复杂的几何体，以此展示球体体积的推导过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。